All Questions ⁺⁰236007 Questions

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+2653

Number Theory

You have a total supply of $1000$ pieces of candy, and an empty vat. You also have a machine that can add exactly $52$ pieces of candy per scoop to the vat, and another machine that can remove exactly $39$ pieces of candy with a different scoop from read more ..

LiIIiam0216 Aug 5, 2024

0

16

0

+598

Number Theory

When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.
When read more ..

cooIcooIcooI17 Aug 5, 2024

+1

14

1

+24

Let $m$ and $n$ be positive integers. If $m$ has exactly $7$ positive divisors, $n$ has exactly $10$ positive divisors, and $mn$ has exactly

Let $m$ and $n$ be positive integers. If $m$ has exactly $7$ positive divisors, $n$ has exactly $10$ positive divisors, and $mn$ has exactly $22$ positive divisors, then how many divisors does $m^2 n^2$ have?

dogsrule Aug 5, 2024

Aug 4, 2024

0

12

0

+598

Number Theory

Let $a$, $b$, $c$, and $n$ be positive integers. If $a + b + c = 19 \cdot 97$ and
a + 3n = b - 2n = \frac{c}{5n},
compute the value of $a$.

cooIcooIcooI17 Aug 4, 2024

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12

0

+598

Number Theory

Find the largest positive integer n such that
\frac{(n + 1)^2}{n + 2}
is an integer.

cooIcooIcooI17 Aug 4, 2024

0

13

0

+1234

Number Theory

On planet Enigma, the residents use a currency called the confusion. There are only 2 confusion bills on Enigma, one worth 2 confusions and the other worth 3 confusions. There are also some coins of smaller value, but each weighs over 10 read more ..

Akhain1 Aug 4, 2024

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13

1

+2653

Number Theory

The product of a set of distinct positive integers is $630$. If one of the numbers is 42, what are the other two numbers?

LiIIiam0216 Aug 4, 2024

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13

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+1839

Number Theory

How many positive four-digit integers $N$ have the property that the three-digit number obtained by removing the leftmost digit is equal to $\frac{N}{24}?$

ABJeIIy Aug 4, 2024

0

10

1

+948

Geometry

In right triangle $ABC,$ $\angle C = 90^\circ$. Median $\overline{AM}$ has a length of 1, and median $\overline{BN}$ has a length of 1. What is the length of the hypotenuse of the triangle?

Rangcr897 Aug 4, 2024

+2

10

1

+19

Probability

Three fair quarters are tossed, and a tail appears on at least one of them. What is the probability that at least one head appears? Express your answer as a common fraction.

DeluluSunshine Aug 4, 2024

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10

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+842

Algebra

Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible (that is, $f^{-1}(x)$ exists for all $x$ in the range of $f$). If the graphs of $y=f(x)$ and $y=f(x^3)$ are drawn, at how many points do they intersect?

RedDragonl Aug 4, 2024

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12

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+842

Algebra

Let $A$, $B$, and $C$ be constants so that
\frac{1}{x^3-2x^2-5x+9}=\frac{A}{x+5}+\frac{B}{x-3} + \frac{C}{(x - 3)^2}
holds for all real numbers $x$ other than $-5$ and $3.$ What is $A$?

RedDragonl Aug 4, 2024

0

13

1

+842

Algebra

The parabola $y = ax^2 + bx + c$ is graphed below. Find $a \cdot b \cdot c.$ (The grid lines are one unit apart.) read more ..

RedDragonl Aug 4, 2024

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12

1

+842

Algebra

The equation of the line below can be expressed in the form
\frac{x}{a} + \frac{y}{b} = 1.
Find a. read more ..

RedDragonl Aug 4, 2024

0

4

1

+948

Geometry

In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

Rangcr897 Aug 4, 2024

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12

1

+663

Algebra

Find the discriminant of the quadratic 5x^2 - 2x + 8 - 2x^2 + 6x + 2.

siIviajendeukie Aug 4, 2024

0

9

1

+663

Algebra

Find the largest integer k such that the equation 5x^2 - kx + 8 - 2x^2 + 25 =0 has no real solutions

siIviajendeukie Aug 4, 2024

0

10

1

+2653

Algebra

The quadratic equation $x^2-mx+24 = 10$ has roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?

LiIIiam0216 Aug 4, 2024

0

12

1

+2653

Algebra

Evaluate the infinite geometric series
0.4 + 0.036 + 0.0000324 + ...
Express your answer as a fraction with integer numerator and denominator.

LiIIiam0216 Aug 4, 2024

0

9

1

+2653

Algebra

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 2 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator read more ..

LiIIiam0216 Aug 4, 2024

0

12

1

+785

help algebra

The solutions to
2x^2 - 10x + 13 = -6x^2 - 18x - 15
are a+bi and a-bi, where a and b are positive. What is a\cdot b?

gnistory Aug 4, 2024

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